FDCL Part V: Gauge Hamiltonians and Scattering on FDCL Graphs — Exact Reductions and Observable Quotients
- 1Independent Researcher
Abstract
We study spectral reduction on explicitly defined recursive graphs, retaining the mass, boundary degrees and observations that determine each comparison. For the ordinary six-corner voxel Laplacian with unit conductances and counting mass, explicit currents and a stopped-coordinate trial prove 30/[11(6^n-1)] <= Delta_n <= (80/7)6^(-n). With N=6^n, its eigenvalue counting function is comparable to N*lambda on an explicit finite-level spectral window. This corresponds to counting exponent d_s=2 on the specified finite-level window. A continuum spectral dimension or heat-kernel interpretation requires a limiting Dirichlet form. Consequences include an averaged return window of order t^(-1), logarithmic mean effective resistance and eigenvalue order j^(-1) for every normalized Green cluster. An indexed harmonic Ritz enclosure has an optimal universal scalar function with leading small-error coefficient 1/4. Refining the actual cell means gives relative error O_j(6^(-2k)); at most 36 retained cells or vertices guarantee gap error below one percent at every level. The fixed-six relative error remains positive, and both least positive modes are simple in the same symmetry sector for n>=11. For an ordered bowtie hierarchy, local pair modes exhaust the ground space from level four. Direct routes prove the uniform excitation gap 7/428300; an exact finite certificate improves it to 609/1602436. Each of two ordered six-vertex signless families has lower bound 1/3510 for every n>=3, and its free and boundary-corrected ground energies converge to a common positive limit. We give seed-dependent quantitative bounds and determine the limiting energy for fixed positive and negative boundary loads. Growing-dimensional subspaces near the unloaded ground energy retain positive outer mass. Fixed-background unitary connections, mass-aware passive responses, rational bond scattering and two finite observable quotients provide further reductions with stated model data. The ratio-limit analysis and its conditional criteria are collected in appendices; actual full-sequence ratio convergence, its exact limit and the full unloaded ground-state boundary mass remain open. The article includes analytic proofs and the exact inputs needed for its computer-assisted finite statements. Version 2.0 is an integrated update of FDCL Part V, previously archived as version 1.0 at https://doi.org/10.5281/zenodo.23227317. The manuscript is dated 23 October 2026 (134 pages). The archive contains 727 files: the article, editable LaTeX sources, fourteen used vector figures, exact scientific inputs, retained certificates and verification programs. The bibliography has 116 entries. External reference PDFs and internal editorial or review records are excluded. Open infinite-generation questions and the scope of the finite certificates are stated explicitly. This is a working paper, not a peer-reviewed publication. Generative AI, including GPT models and Codex (OpenAI) and Claude models (Anthropic), was used substantively in derivations, proof development, computational analysis, verification code, literature comparison and drafting. The author formulated and directed the research, evaluated the material and accepts responsibility for the mathematical statements, interpretations, references and final manuscript. AI assistance is disclosed separately from the analytic proofs and exact finite certificates.
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BibTeX
@article{seol2026fdcl,
title = {{FDCL Part V: Gauge Hamiltonians and Scattering on FDCL Graphs — Exact Reductions and Observable Quotients}},
author = {Bin Seol},
journal = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23227316},
url = {https://doi.org/10.5281/zenodo.23227316},
note = {CC BY},
}
RIS
TY - JOUR
TI - FDCL Part V: Gauge Hamiltonians and Scattering on FDCL Graphs — Exact Reductions and Observable Quotients
AU - Bin Seol
PY - 2026
DA - 2026/10/08
JO - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.23227316
UR - https://doi.org/10.5281/zenodo.23227316
AB - We study spectral reduction on explicitly defined recursive graphs, retaining the mass, boundary degrees and observations that determine each comparison. For the ordinary six-corner voxel Laplacian with unit conductances and counting mass, explicit currents and a stopped-coordinate trial prove 30/[11(6^n-1)] <= Delta_n <= (80/7)6^(-n). With N=6^n, its eigenvalue counting function is comparable to N*lambda on an explicit finite-level spectral window. This corresponds to counting exponent d_s=2 on the specified finite-level window. A continuum spectral dimension or heat-kernel interpretation requires a limiting Dirichlet form. Consequences include an averaged return window of order t^(-1), logarithmic mean effective resistance and eigenvalue order j^(-1) for every normalized Green cluster. An indexed harmonic Ritz enclosure has an optimal universal scalar function with leading small-error coefficient 1/4. Refining the actual cell means gives relative error O_j(6^(-2k)); at most 36 retained cells or vertices guarantee gap error below one percent at every level. The fixed-six relative error remains positive, and both least positive modes are simple in the same symmetry sector for n>=11. For an ordered bowtie hierarchy, local pair modes exhaust the ground space from level four. Direct routes prove the uniform excitation gap 7/428300; an exact finite certificate improves it to 609/1602436. Each of two ordered six-vertex signless families has lower bound 1/3510 for every n>=3, and its free and boundary-corrected ground energies converge to a common positive limit. We give seed-dependent quantitative bounds and determine the limiting energy for fixed positive and negative boundary loads. Growing-dimensional subspaces near the unloaded ground energy retain positive outer mass. Fixed-background unitary connections, mass-aware passive responses, rational bond scattering and two finite observable quotients provide further reductions with stated model data. The ratio-limit analysis and its conditional criteria are collected in appendices; actual full-sequence ratio convergence, its exact limit and the full unloaded ground-state boundary mass remain open. The article includes analytic proofs and the exact inputs needed for its computer-assisted finite statements. Version 2.0 is an integrated update of FDCL Part V, previously archived as version 1.0 at https://doi.org/10.5281/zenodo.23227317. The manuscript is dated 23 October 2026 (134 pages). The archive contains 727 files: the article, editable LaTeX sources, fourteen used vector figures, exact scientific inputs, retained certificates and verification programs. The bibliography has 116 entries. External reference PDFs and internal editorial or review records are excluded. Open infinite-generation questions and the scope of the finite certificates are stated explicitly. This is a working paper, not a peer-reviewed publication. Generative AI, including GPT models and Codex (OpenAI) and Claude models (Anthropic), was used substantively in derivations, proof development, computational analysis, verification code, literature comparison and drafting. The author formulated and directed the research, evaluated the material and accepts responsibility for the mathematical statements, interpretations, references and final manuscript. AI assistance is disclosed separately from the analytic proofs and exact finite certificates.
ER -